Bridge mixtures of random walks on an Abelian group

In this paper, we characterize (mixtures of) bridges of a continuous time random walk with values in a countable Abelian group. Our main tool is a conditional version of Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of such random processes. We observe in particular how their structure depends on the algebraic properties of the underlying group.